Theorem 2mos

Description: Double "there exists at most one", using implicit substitution. (Contributed by NM, 10-Feb-2005) (Proof shortened by Wolf Lammen, 21-May-2025)

	Ref	Expression
Hypothesis	2mos.1	⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → ( 𝜑 ↔ 𝜓 ) )
Assertion	2mos	⊢ ( ∃ 𝑧 ∃ 𝑤 ∀ 𝑥 ∀ 𝑦 ( 𝜑 → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) )

Proof

Step	Hyp	Ref	Expression
1		2mos.1	⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → ( 𝜑 ↔ 𝜓 ) )
2		2mo	⊢ ( ∃ 𝑧 ∃ 𝑤 ∀ 𝑥 ∀ 𝑦 ( 𝜑 → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) )
3	1	2sbievw	⊢ ( [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ↔ 𝜓 )
4	3	anbi2i	⊢ ( ( 𝜑 ∧ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ) ↔ ( 𝜑 ∧ 𝜓 ) )
5	4	imbi1i	⊢ ( ( ( 𝜑 ∧ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) )
6	5	2albii	⊢ ( ∀ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) ↔ ∀ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) )
7	6	2albii	⊢ ( ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) )
8	2 7	bitri	⊢ ( ∃ 𝑧 ∃ 𝑤 ∀ 𝑥 ∀ 𝑦 ( 𝜑 → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) )